>as n increases, pieces are removed from the board for example restricting the number of legal moves //
As the pawns are moved to make way for the back row the number of legal moves increases very quickly. For example if white plays d2-d3 then it frees the King, Queen and Bishop to move; 2 possible moves are removed and 8 are added.
Right, but eventually the number of possible next moves starts to drop. A game with a King, Queen, and Two Bishops has more possible valid next moves than a game with just a King and a Bishop left.
As the pawns are moved to make way for the back row the number of legal moves increases very quickly. For example if white plays d2-d3 then it frees the King, Queen and Bishop to move; 2 possible moves are removed and 8 are added.